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Theorem elsymrelsrel 38539
Description: For sets, being an element of the class of symmetric relations (df-symrels 38525) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
elsymrelsrel (𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))

Proof of Theorem elsymrelsrel
StepHypRef Expression
1 elrelsrel 38469 . . 3 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
21anbi2d 630 . 2 (𝑅𝑉 → ((𝑅𝑅𝑅 ∈ Rels ) ↔ (𝑅𝑅 ∧ Rel 𝑅)))
3 elsymrels2 38535 . 2 (𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))
4 dfsymrel2 38531 . 2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
52, 3, 43bitr4g 314 1 (𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wss 3963  ccnv 5688  Rel wrel 5694   Rels crels 38164   SymRels csymrels 38173   SymRel wsymrel 38174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-rels 38467  df-ssr 38480  df-syms 38524  df-symrels 38525  df-symrel 38526
This theorem is referenced by:  elrefsymrelsrel  38553
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